Coordinate representation for non Hermitian position and momentum operators

Abstract

In this paper we undertake an analysis of the eigenstates of two non self-adjoint operators $\hat{q}$ and $\hat{p}$ similar, in a suitable sense, to the self-adjoint position and momentum operators ${\hat{q}}_0$ and ${\hat{p}}_0$ usually adopted in ordinary quantum mechanics. In particular we discuss conditions for these eigenstates to be {\em biorthogonal distributions}, and we discuss few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with $\hat{q}$ and $\hat{p}$, based on the so-called {\em quasi *-algebras}.

arXiv commentAccepted in Proceedings of the Royal Society A

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Cited by in corpus (6)

• Biorthogonal vectors, sesquilinear forms and some physical operators
• Weak pseudo-bosons
• Bi-squeezed states arising from pseudo-bosons
• Tridiagonality, supersymmetry and non self-adjoint Hamiltonians
• Bi-coherent states as generalized eigenstates of the position and the momentum operators
• Coupled Susy, pseudo-bosons and a deformed $\mathfrak{su}(1,1)$ Lie algebra